Hatsuyume

The first dream a person has in the new year. The contents of the dream usually foretells the luck of the dreamer, and it is considered a sign of particularly good luck to dream of a hawk, an eggplant, and Mt. Fuji. Note: this is the dream you have on New Year's Eve night. If you don't remember your hatsuyume, then you don't have one.

Examples of Hatsuyume include:

Anime and Manga

Azumanga Daioh dedicates a entire episode to the different dreams of the Osaka, Tomo, Sakaki and Kaorin, though only Sakaki dreams of the aforementioned hawk, eggplant and Mt. Fuji.
- All delivered by Osaka, of course.
The first episode of Haruhi-chan portays the events of Kyons dream, with Koizumi wearing a Mt. Fuji costume (which turns out be just a random mountain), Tsuruya in a crane costume (who then claims to have the heart of a hawk) and Yuki as a eggplant cellphone strap. It's that kind of show.
- The manga has a second one where Kyon's Sister becomes a cow.
In ×××HOLiC, Watanuki's Hatsuyume-induced good mood quickly evaporates when, once again, he becomes the Butt Monkey of Yuko.
Lucky Star has Miyuki point out that the list follows on with a fan, tobacco, and a blind man. Konata wonders who dreams about these sorts of things anyway, what sort of dream will include all these elements, and how many people remember what they dream about that clearly.
In Digimon Adventure, one of the mini-dramas was called "Nusumareta Hatsuyume," or "Stolen New Year's Dreams." From a not-really fluent listen-through of it, basically, some Digimon eats everyone in Tokyo's first dreams of the New Year and Takeru and Yamato somehow defeat the digimon after a heck of a lot of bizarre sound effects.
An episode of Saber Marionette J has Hanagata finding proof everywhere that he should propose to Otaru. This leads to having a dream about Mt. Fuji, a hawk, an eggplant...and Otaru.
In Cardcaptor Sakura, Sakura Kinomoto transforms The Dream Clow Card into a Sakura Card during her hatsuyume. She also has a vision of Eriol and his guardians while at it.

Video Games

Referenced in Parodius da!: the volcanoes in the Wutai stage shoot eggplants at you.
As an Easter Egg, some Harvest Moon games have several extremely bizarre dreams you can have if you go to bed late on the last day of the year.
In LSD Dream Emulator, the 365th dream in the game ends with a cutscene featuring Mt Fuji, a hawk, and an eggplant.